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High Rate Efficient Local List Decoding from HDX

Yotam Dikstein, Max Hopkins, Russell Impagliazzo, Toniann Pitassi

TL;DR

The paper introduces approximate locally list-decodable codes (aLLDCs) with rate, efficiency, and error tolerance approaching information-theoretic limits, built on high dimensional expanders (HDX). It develops two HDX-based frameworks: (i) sub-constant error aLLDCs using polylog-round belief propagation and strongly explicit routing, and (ii) constant-rate aLLDCs using a three-layer hypergraph system (V,S,U) with inner/outer decoding and fault-tolerant routing. The constructions yield polylog-rate codes decodable in polylog-depth with near-optimal query complexity, and constant-rate aLLDCs enabling list-decoding in RNC^1-style settings when combined with classical outer codes. Applications span hardness amplification, near-optimal fast PRGs, complexity-preserving distance amplification, and the creation of high-rate LLDCs, marking significant progress toward practical, locally decodable, high-rate codes. The work also establishes lower bounds showing near-polynomial gaps to optimality and maps a detailed circuit-implementation program for the HDX-based aLLDCs, with a comprehensive treatment of HDX-based routing, sampling, and coset encodings.

Abstract

We construct the first (locally computable, approximately) locally list decodable codes with rate, efficiency, and error tolerance approaching the information theoretic limit, a core regime of interest for the complexity theoretic task of hardness amplification. Our algorithms run in polylogarithmic time and sub-logarithmic depth, which together with classic constructions in the unique decoding (low-noise) regime leads to the resolution of several long-standing problems in coding and complexity theory: 1. Near-optimally input-preserving hardness amplification (and corresponding fast PRGs) 2. Constant rate codes with $\log(N)$-depth list decoding (RNC$^1$) 3. Complexity-preserving distance amplification Our codes are built on the powerful theory of (local-spectral) high dimensional expanders (HDX). At a technical level, we make two key contributions. First, we introduce a new framework for ($\mathrm{polylog(N)}$-round) belief propagation on HDX that leverages a mix of local correction and global expansion to control error build-up while maintaining high rate. Second, we introduce the notion of strongly explicit local routing on HDX, local algorithms that given any two target vertices, output a random path between them in only polylogarithmic time (and, preferably, sub-logarithmic depth). Constructing such schemes on certain coset HDX allows us to instantiate our otherwise combinatorial framework in polylogarithmic time and low depth, completing the result.

High Rate Efficient Local List Decoding from HDX

TL;DR

The paper introduces approximate locally list-decodable codes (aLLDCs) with rate, efficiency, and error tolerance approaching information-theoretic limits, built on high dimensional expanders (HDX). It develops two HDX-based frameworks: (i) sub-constant error aLLDCs using polylog-round belief propagation and strongly explicit routing, and (ii) constant-rate aLLDCs using a three-layer hypergraph system (V,S,U) with inner/outer decoding and fault-tolerant routing. The constructions yield polylog-rate codes decodable in polylog-depth with near-optimal query complexity, and constant-rate aLLDCs enabling list-decoding in RNC^1-style settings when combined with classical outer codes. Applications span hardness amplification, near-optimal fast PRGs, complexity-preserving distance amplification, and the creation of high-rate LLDCs, marking significant progress toward practical, locally decodable, high-rate codes. The work also establishes lower bounds showing near-polynomial gaps to optimality and maps a detailed circuit-implementation program for the HDX-based aLLDCs, with a comprehensive treatment of HDX-based routing, sampling, and coset encodings.

Abstract

We construct the first (locally computable, approximately) locally list decodable codes with rate, efficiency, and error tolerance approaching the information theoretic limit, a core regime of interest for the complexity theoretic task of hardness amplification. Our algorithms run in polylogarithmic time and sub-logarithmic depth, which together with classic constructions in the unique decoding (low-noise) regime leads to the resolution of several long-standing problems in coding and complexity theory: 1. Near-optimally input-preserving hardness amplification (and corresponding fast PRGs) 2. Constant rate codes with -depth list decoding (RNC) 3. Complexity-preserving distance amplification Our codes are built on the powerful theory of (local-spectral) high dimensional expanders (HDX). At a technical level, we make two key contributions. First, we introduce a new framework for (-round) belief propagation on HDX that leverages a mix of local correction and global expansion to control error build-up while maintaining high rate. Second, we introduce the notion of strongly explicit local routing on HDX, local algorithms that given any two target vertices, output a random path between them in only polylogarithmic time (and, preferably, sub-logarithmic depth). Constructing such schemes on certain coset HDX allows us to instantiate our otherwise combinatorial framework in polylogarithmic time and low depth, completing the result.
Paper Structure (149 sections, 61 theorems, 125 equations, 14 figures, 2 tables)

This paper contains 149 sections, 61 theorems, 125 equations, 14 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Motivation and Background
  3. Hardness Amplification
  4. Complexity Preserving Distance-Amplification
  5. List Decodable Codes with Low-depth Encoding and Decoding
  6. Locally List Decodable Codes
  7. Main Results: High Rate aLLDCs
  8. Applications in Complexity and Coding Theory
  9. Local and Parallel List-Decoding
  10. Hardness Amplification and Fast PRGs
  11. Summary of aLLDC Parameters
  12. Proof Overview
  13. aLLDCs with Sub-Constant Error (\ref{['thm:intro-intro-polylog']})
  14. Reduction to List Recovery
  15. List recovery setup
  16. ...and 134 more sections

Key Result

Theorem 1.1

For every $N \in \mathbb{N}$ and $\varepsilon \leq \frac{1}{\log(N)}$, there is a binary aLLDC $\mathcal{C}_N$ decodable from $\frac{1}{2}-\varepsilon$ errors with: Moreover $\mathcal{C}_N$ is decodable in $\mathop{\mathrm{poly}}\nolimits(\frac{1}{\varepsilon})$-size and $\tilde{O}(\log^2\frac{1}{\varepsilon})$-depth, and encodable in time $\mathop{\mathrm{poly}}\nolimits(\frac{1}{\varepsilon})$.

Figures (14)

  • Figure 1: A Decoding Path
  • Figure 2: List-recovery algorithm
  • Figure 3: The Subspace Set System
  • Figure 4: Inner Decoder. \ref{['claim:const-rate-inner-decoder']} Follows by outputting $InDec_{\delta_{in}}(s,\cdot)$ for $O(\frac{1}{\varepsilon})$ random choices of $s \in S_u$. See \ref{['app:inner-code']} for formal proof.
  • Figure 5: Outer decoder circuit algorithm
  • ...and 9 more figures

Theorems & Definitions (174)

  • Theorem 1.1: Polylog Rate aLLDCs (\ref{['cor:binary-polylog-aLLDCs']})
  • Theorem 1.2: Constant Rate aLLDCs (\ref{['cor:constant-rate-binary']})
  • Corollary 1.3: LLDCs in sub-Polynomial Time (\ref{['thm:sub-Poly-LLDCs']})
  • Corollary 1.4: List Decoding in RNC$^1$ (\ref{['thm:constant-rate-NC']})
  • Theorem 1.5: Uniform complexity-preserving hardness amplification (\ref{['thm:ucpha']})
  • Theorem 1.6: Non-uniform complexity preserving hardness amplification (\ref{['thm:nucpha']})
  • Theorem 1.7: Near-Optimal Fast PRGs (\ref{['thm:PRG']})
  • Definition 2.1
  • Definition 2.2: Approximate Locally List-Decodable Codes (aLLDCs)
  • Definition 2.3: Hypergraph System
  • ...and 164 more